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Feb 1, 2012 at 9:40 comment added Jochen Wengenroth In Ralf's question, $V_2$ is nuclear whenever $V_1$ and $V_3$ are nuclear. This can be reformulated without exact sequences: A locally convex space is nuclear whenever it has a nuclear subspaces such that the corresponding quotient is also nuclear. In Dierolf's and Roelcke's article there many more results of this type as well as many counterexamples.
Feb 1, 2012 at 9:28 comment added Andrew Stacey But answered which way? Could you add a brief note at least saying what the answer actually is so that those casually reading this, or those without access to the references, can at least know what the answer is.
Feb 1, 2012 at 9:02 history answered Jochen Wengenroth CC BY-SA 3.0